Necessary conditions for two polynomials to have the same set of real roots

[annarodriguez]

If P(X) and Q(X) ∈ R[X] and P(X) | P( Q(X) ), what should be the necessary conditions for Q(X) such that the set of the real roots of P(X) to be equal to the set of the real roots of Q(X)−X ( i.e. the set of fixed points of the polynomial function of Q(X) ) ?

 

[Steven G]

Nice problem. Did you read the guidelines that ask you to tell us where you are stuck so we know what kind of help you need? We hate showing something to a student that they already know as it is not very productive, So please show us the work you have done so far, even if you know its wrong.

 

[annarodriguez]

Nice problem. Did you read the guidelines that ask you to tell us where you are stuck so we know what kind of help you need? We hate showing something to a student that they already know as it is not very productive, So please show us the work you have done so far, even if you know its wrong.

Sorry, here's my work so far:

I've started from a particular case, i.e. Q(X)=X². More specific, there was this problem asking to find all real monic polynomials P(X), with simple real roots, such that P(X^2) = ± P(X) * P(-X). I've found out that the only such possible polynomials are P(X)=X, P(X)=X-1 or P(X)=X(X-1)=X²-X.

I was trying to find a more general approach to this particular problem, but the only incomplete result I've managed to find by now is this one:
If the polynomial function Q is strictly monotone on the interval Im(Q), then the set of the real roots of P(X) is included into the set of the real roots of Q(X)-X ( returning to the particular case Q(X)=X², we see that it is strictly increasing on the interval Im(Q)=[0,∞) ).

 

[annarodriguez]

Sorry, here's my work so far:

I've started from a particular case, i.e. Q(X)=X². More specific, there was this problem asking to find all real monic polynomials P(X), with simple real roots, such that P(X^2) = ± P(X) * P(-X). I've found out that the only such possible polynomials are P(X)=X, P(X)=X-1 or P(X)=X(X-1)=X²-X.

I was trying to find a more general approach to this particular problem, but the only incomplete result I've managed to find by now is this one:
If the polynomial function Q is strictly monotone on the interval Im(Q), then the set of the real roots of P(X) is included into the set of the real roots of Q(X)-X ( returning to the particular case Q(X)=X², we see that it is strictly increasing on the interval Im(Q)=[0,∞) ).

One necessary condition would also be that Q does not induce a permutation without fixed points on any finite subset of ℝ, i.e. there does not exist a finite set S⊂ℝ such that Q(S)=S but Q(s)≠s for all s∈ S. Namely, if such S existed we could take P(X)=∏_{s∈ S}(X−s).
However, it is still only an incomplete result, and I got quite stuck from here...